Determining the Altitudes of a Triangle with Sides 13, 14, and 15

In this article, we will explore the process of determining the lengths of the altitudes of a triangle with side lengths of 13, 14, and 15. We will use both Heron's formula to find the area of the triangle and the relationship between the area and the altitudes to find the altitudes.

Introduction

Altitudes of a triangle are perpendicular lines drawn from each vertex to the opposing side. The lengths of the altitudes are crucial in understanding the characteristics and properties of a triangle. In this case, we will calculate the altitudes for a triangle with sides 13, 14, and 15.

Step 1: Calculate the Area of the Triangle using Heron's Formula

To find the area of a triangle with sides of lengths 13, 14, and 15, we can use Heron's formula. Heron's formula states that the area (A) of a triangle with sides (a), (b), and (c) is given by:

[A sqrt{s(s-a)(s-b)(s-c)}]

where (s) is the semi-perimeter of the triangle:

[s frac{a b c}{2}]

Given the side lengths (a 13), (b 14), and (c 15), we can calculate the semi-perimeter:

[s frac{13 14 15}{2} 21]

Now, using the semi-perimeter, we can calculate the area:

[A sqrt{21(21-13)(21-14)(21-15)} sqrt{21 times 8 times 7 times 6}]

Let's simplify the expression under the square root:

[21 times 8 168] [7 times 6 42] [168 times 42 7056]

[A sqrt{7056} 84]

Step 2: Calculate the Altitudes

Once we have the area of the triangle, we can use the formula for the altitude corresponding to one side. The altitude (h_a) corresponding to side (a) can be calculated using the formula:

[h_a frac{2A}{a}]

Using this formula, we will calculate the altitudes for each side of the triangle:

Altitude on Side 13

[h_a frac{2 times 84}{13} approx 12.92]

Altitude on Side 14

[h_b frac{2 times 84}{14} 12]

Altitude on Side 15

[h_c frac{2 times 84}{15} approx 11.2]

Summary of the Altitudes

Altitude to side 13: (h_a approx 12.92) Altitude to side 14: (h_b 12) Altitude to side 15: (h_c approx 11.2)

Therefore, the lengths of the three altitudes are approximately:

(h_a approx 12.92) (h_b 12) (h_c approx 11.2)

Conclusion

In summary, we calculated the altitudes of a triangle with given side lengths of 13, 14, and 15 by first finding the area using Heron's formula and then using the relationship between the area and the altitudes to determine the altitudes. This method provides a systematic way to find the altitudes of any triangle when the side lengths are known.